Final answer:
To find the volume of the solid with square cross sections perpendicular to the x-axis, we can integrate the area of the squares along the x-axis. Similarly, to find the volume of the solid with square cross sections perpendicular to the y-axis, we can integrate the area of the squares along the y-axis.
Step-by-step explanation:
The volume of a solid with cross sections perpendicular to the x-axis are squares can be found by finding the area of each square cross section and then integrating along the x-axis. The given base is a circle with equation x^2 + y^2 = 16. To find the area of each square cross section, we need to find the length of one side of the square. This can be done by finding the length of the chord that passes through the center of the circle and is perpendicular to the x-axis. The length of this chord can be found using the Pythagorean theorem. Then, we can integrate the area of the square cross sections along the x-axis from the leftmost point of the circle to the rightmost point of the circle to find the volume of the solid.
Similarly, to find the volume of a solid with cross sections perpendicular to the y-axis that are squares, we need to find the length of one side of the square. This can be done by finding the length of the chord that passes through the center of the circle and is perpendicular to the y-axis. Then, we can integrate the area of the square cross sections along the y-axis from the bottommost point of the circle to the topmost point of the circle to find the volume of the solid.